On irredundance coloring and irredundance compelling coloring of graphs

Abstract

An irredundance coloring of a graph $G$ is a proper coloring admitting a maximal irredundant set all of whose vertices receive different colors. The minimum number of colors required for an irredundance coloring of $G$ is called the irredundance chromatic number of $G$, and is denoted by ${\chi }_i\left(G\right)$. An irredundance compelling coloring of $G$ is a proper coloring of $G$ in which every rainbow committee (a set consisting of one vertex of each color) is an irredundant set of $G$. The maximum number of colors required for an irredundance compelling coloring of $G$ is called the irredundance compelling chromatic number of $G$, and is denoted by ${\chi }_{irc}\left(G\right)$. We make a detailed study of ${\chi }_i\left(G\right)$, ${\chi }_{irc}\left(G\right)$, derive bounds on these parameters and characterize extremal graphs attaining the bounds.